1) Newton's law of viscosity was generalized to account for complex fluid flow patterns where velocity can vary in different directions and over time. 2) By analyzing forces on a small volume element sliced in different orientations, it was determined that stresses have both pressure and viscous components. 3) The viscous stresses were shown to be linear combinations of the velocity gradients, and for isotropic fluids can be expressed by two scalar coefficients multiplied by the symmetric parts of the velocity gradient tensor.

1. Transport Phenomena

2. GENERALIZATION OF
NEWTON'S LAW OF VISCOSITY

3. In this presentation
We explain briefly the main ideas that led to the discovery of the required
generalization of Newton's law of viscosity.

4. • Consider a very general flow pattern, in which the fluid velocity may be in
various directions at various places and may depend on the time t.
• The velocity components are given as

5. • In such a situation, there will be nine stress components τij (where i and j
may take on the designations x, y, and z).
• In figure is shown a small cube-shaped volume element within the flow field
each face having unit area. The center of the volume element is at the
position x, y, z.

6. Figure

7. • At any instant of time we can slice the volume element in such a way as to
remove half the fluid within it. As shown in the figure, we can cut the
volume perpendicular to each of the three coordinate directions in turn. We
can then ask what force has to be applied on the free (shaded) surface in
order to replace the force that had been exerted on that surface by the fluid
that was removed. There will be two contributions to the force: that
associated with the pressure, and that associated with the viscous forces.

8. Figures
• Pressure and viscous forces acting on planes in the fluid perpendicular to the three coordinate systems.
The shaded planes have unit area.
a b c

9. • The pressure force will always be perpendicular to the exposed surface.
Hence in (a) the force per unit area on the shaded surface will be a vector
pδx, that is the pressure (a scalar) multiplied by the unit vector δx, in the x
direction. Similarly, the force on the shaded surface in (b) will be pδy, and in
(c) the force will be pδz. The pressure forces will be exerted when the fluid
is stationary as well as when it is in motion.

10. • The viscous forces come into play only when there are velocity gradients
within the fluid. In general they are neither perpendicular to the surface
element nor parallel to it but rather at some angle to the surface as shown in
figure.
• In (a) we see a force per unit area τx, exerted on the shaded area, and in (b)
and (c) we see forces per unit area and .Each of these forces (which are
vectors) has components (scalars); for example, T, has components T,,, T,~,
and T,,. Hence we can now summarize the forces acting on the thre shaded
areas in Fig. 1.2-1 in Table 1.2-1.

11. • This tabulation is a summary of the forces per unit area (stresses) exerted
within a fluid, both by the thermodynamic pressure and the viscous stresses
• Sometimes we will find it convenient to have a symbol that includes both
types of stresses, and so we define the molecular stresses as follows:

12. • πij =ρδ ij +τ ij : force in the j direction on a unit area perpendicular to the i
direction, where it is understood that the fluid in the region of lesser xi is
exerting the force on the fluid of greater xi
• πij =ρδ ij +τ ij : flux of j-momentum in the positive i direction-that is, from
the region of lesser xi to that of greater xi
• Data is represented in the next table

14. • The viscous stresses may be linear combinations of all the velocity gradients
• We assert that time derivatives or time integrals should not appear in the
expression

15. • We do not expect any viscous forces to be present, if the fluid is in a state of
pure rotation. This requirement leads to the necessity that τi, be a symmetric
combination of the velocity gradients. By this we mean that if i and j are
interchanged, the combination of velocity gradients remains unchanged. It
can be shown that the only symmetric linear combinations of velocity
gradients are

16. • If the fluid is isotropic-that is, it has no preferred direction-then the
coefficients in front of the two expressions in Eq must be scalars so that
• Of course, we will simplify Eq. for the flow situation in Fig. For that
elementary flow Eq. simplifies to τ = A dv,/dy, and hence the scalar constant
A must be the same as the negative of the viscosity

17. • Finally, by common agreement among most fluid dynamicists the scalar
constant B is set equal to μ - K
• where K is called the dilatational viscosity. The reason for writing B in this
way is that it is known from kinetic theory that K is identically zero for
monatomic gases at low density

18. • Thus the required generalization for Newton's law of viscosity in Eq. 1.1-2 is
then the set of nine relations (six being independent):
• this set of relations can be written more concisely in the vector-tensor
notation